In this course, one continues the study of derivatives and integrals begun in Calculus I & Calculus II. However, the perspective is broadened in several ways:
This broadening of the point of view has several practical consequences, paramount among them is the fact that the graphs of such functions no longer fit in a plane - more dimensions are needed. For example a function z = f(x,y) has a graph which is a surface in 3-space, if there are more independent variables, even more dimensions are required, making it difficult or impossible to visualize. Similarly, vector-valued functions usually require more dimensions, so an alternate way of visualizing them, known as vector-fields (particularly useful in Physics - electromagnetic fields are the classic example) are sometimes employed.
Another wrinkle is that in calculus I when computing limits, such as the limit of f(x) as x approaches a, one need only consider x approaching a from the left or from the right. When approaching a point in higher dimensions however, one can approach from infinitely many different directions; even worse, one can approach a point along a straight line or along some curved path, even one which is quite convoluted, spiralling around the target point for example! This makes it more difficult for limits to exist, and special care must be given to any definition which depends on the notion of a limit. This means we need to take a careful look at what it means for a function of several variables to be continuous, to be differentiable, to be integrable. We will discover some new twists in higher dimensions which do not occur in the more restricted situation of Calculus I & II.
In the end however, all the usual notions of Calculus will extend to the multivariable case, but usually with more variety in the possibilities. For example, in place of the derivative of a function, one has the partial derivatives (note the plural!), the directional derivative, and the total derivative (also known as the gradient). On the integration side, we have iterated integrals, double interals (or triple integrals, etc.), potential functions (something like an antiderivative), line integrals, and surface integrals. All of these concepts have their uses (some of which will look quite familiar - for example there is a way to use the gradient of a function to find the maximum and minimum values of a function, and a second derivative test to distinguish between them, just as in Calculus I. But the real beauty in this material is in the new power it gives the calculus to handle applications unavailable in Calculus I or II, and also in the inter-relationships between these new concepts. In fact our goal will be to get to Gauss's theorem, which explores the relationship between a surface integral and a triple integral.
We will achieve these goals by covering Chapters 12 - 16 of our text. At times, it may be useful to look at supplementary material, such as on polar coordinates. Other times it would be useful to see some applications not covered in the text, such as the historical derivation of Kepler's laws of planetary motion from Newton's law of gravity, or more details on vector fields and electro-magnetism. Time permitting, when this material is covered, supplementary notes wil be provided, so that students need purchase only one text.
One last comment: because of the presence of higher dimensions, and because we may consider vector-valued functions, it is seen as an advantage to know some linear algebra. This is not to say that Calculus III cannot be learned by someone ignorant of linear algebra, it is just that everything makes so much more sense when viewed from this perspective. (By the same token, after learning Calculus III, one often gains a better understanding of some of the ideas in linear algebra. The two complement each other.) Nevertheless, MA200 is no longer listed as a pre-requisite for MA202 as it was in the past. We will develop any concepts we need from linear algebra as we go along.